A new method for modelling the stochastic differential equations

Year 2025, Volume: 54 Issue: 6, 2380 - 2398, 30.12.2025

Nihal İnce , Sevil Şentürk

https://doi.org/10.15672/hujms.1637431

Abstract

This study presents a novel approach to estimate the probability density function of solutions to stochastic differential equations using generalized entropy optimization methods. Unlike traditional methods such as the Fokker–Planck–Kolmogorov equation, the proposed generalized entropy optimization methods framework accommodates cases where the distribution of the solution deviates from standard statistical forms. The method integrates the Euler–Maruyama scheme to generate multiple trajectories, producing random variables $\hat{X}(t)$ for each time $t$. The performance of method is evaluated through a comprehensive simulation study, in which it is compared with existing techniques under various parameter settings. Both generalized MaxEnt and MinxEnt distributions are applied, with results indicating that generalized MinxEnt distributions offer superior adaptability and accuracy. Visual and statistical comparisons confirm the theoretical validity and practical efficiency of the method. This framework not only provides a flexible alternative for probability density function estimation in stochastic differential equation modeling but also opens pathways for applications in fuzzy stochastic differential equation systems.

Keywords

Euler-Maruyama method , generalized entropy optimization methods , stochastic differential equations , stochastic process

Ethical Statement

The author declares that has no conflict of interest.

Supporting Institution

This study is supported by the Eskisehir Technical University Scientific Research Projects Commission under grant No. 20DRP046.

Project Number

This study is supported by the Eskisehir Technical University Scientific Research Projects Commission under grant No. 20DRP046

Thanks

We would like to thank Prof. Dr. Aladdin SHAMILOV for the continuous support of knowledge and theoretical support

References

• [1] C. E. Shannon, A mathematical theory of communication, Bell Syst. Tech. J. 27 (3), 379-423, 1948.
• [2] J. Baker-Jarvis, M. Racine and J. Alameddine, Solving differential equations by a maximum entropy-minimum norm method with applications to Fokker-Planck equations, J. Math. Phys. 30 (7), 1459-1463, 1989.
• [3] G. Jumarie, Solution of the multivariate Fokker-Planck equation by using a maximum path entropy principle, J. Math. Phys. 31 (10), 2389-2392, 1990.
• [4] H. P. Langtangen, A general numerical solution method for Fokker-Planck equations with applications structural reliability, Probabilist. Eng. Mech. 6 (1), 1991.
• [5] J. Trebicki and K. Sobczyk, Maximum entropy principle and non-stationary distributions of stochastic systems, Probabilist. Eng. Mech. 11, 169-178, 1996.
• [6] S. A. El-Wakil, A. Elhanbaly and M. A. Abdou, Solution of Fokker-Planck equation by means of maximum entropy approach, J. Quant. Spectrosc. Radiat. Transf. 69, 41-48, 2001.
• [7] C. Soize, Construction of probability distributions in high dimension using the maximum entropy principle: Applications to stochastic processes, random fields and random matrices, Int. J. Numer. Methods Eng. 76 (10), 1583-1611, 2008.
• [8] V. P. Koverda and V. N. Skokov, Maximum entropy and stability of a random process with a 1/f power spectrum under deterministic action, Physica A 391 (23), 5850-5857, 2012.
• [9] W. Zhang, H. Wang, C. Hartmann, M. Weber and C. Schütte, Applications of the cross-entropy method to importance sampling and optimal control diffusions, SIAM J. Sci. Comput. 36 (6), A2654-A2672, 2014.
• [10] M. Opper, An estimator for the relative entropy rate of path measures for stochastic differential equations, J. Comput. Phys. 330, 127-133, 2017.
• [11] G. L. Vasconcelos, D. S. Salazar and A. M. S. Macêdo, Maximum entropy approach to H-theory: Statistical mechanics of hierarchical systems, Phys. Rev. E 97 (2), 022104, 2018.
• [12] V. M. Deshpande and R. Bhattacharya, Data-driven Solution of Stochastic Differential Equations Using Maximum Entropy Basis Functions, IFAC-Pap. Online 53 (2), 7234-7239, 2020.
• [13] B. A. Bhat, A. A. Rather, M. A. K. Baig, D. Qayoom and B. R. Elemary, Weighted Generalized Interval Cumulative Residual Entropy: Properties and its Application, Lobachevskii J. Math. 45 (9), 4069-4080, 2024.
• [14] D. Qayoom, A. A. Rather, N. Alsadat, E. Hussam and A. M. Gemeay, A New Class of Lindley Distribution: System Reliability, Simulation and Applications, Heliyon 10 (19), 1-31, 2024.
• [15] A. A. Rather, M. Azeem, M. Alam, C. Subramanian, G. Ozel and I. Ali, Weighted Erlang-Truncated Exponential Distribution: System Reliability Optimization, Structural Properties, and Simulation, Lobachevskii J. Math. 45 (9), 4311-4337, 2024.
• [16] E. Allen, Modeling with Itô Stochastic Differential Equations, Springer, 2007.
• [17] G. Ozel Kadılar, Stokastik Süreçler ve R Uygulamaları, Seçkin Yayıncılık, Ankara, 2020.
• [18] J. P. Bishwal, Parameter Estimation in Stochastic Differential Equations, Springer, 2007.
• [19] J. Bak, H. Madsen and H. A. Nielsen, Goodness of fit of stochastic differential equations, in Symposium i Anvendt Statistik, 341-346, Copenhagen Business School, Copenhagen, Denmark, 1999.
• [20] T. Sauer, Numerical solution of stochastic differential equations in finance, in Handbook of Computational Finance, Springer, Berlin, Heidelberg, 2012.
• [21] A. Shamilov, A development of entropy optimization methods, WSEAS Trans. Math. 5 (5), 568-575, 2006.
• [22] A. Shamilov, Generalized Entropy Optimization Problems and the Existence of Their Solutions, Physica A 382 (2), 465-472, 2007.
• [23] A. Shamilov, Generalized entropy optimization problems with finite moment function sets, J. Stat. Manag. Syst. 3 (3), 595-603, 2010.
• [24] N. Ince and A. Shamilov, An application of new method to obtain probability density function of solution of stochastic differential equations, Appl. Math. Nonlinear Sci. 5 (1), 337-348, 2020.
• [25] N. Ince, A generalized entropy optimization modelling in the theory of stochastic differential equations, J. Korean Stat. Soc. 1-19, 2021.
• [26] A. Shamilov, To Approximate Distributions of Solutions of Stochastic Differential Equations, Pak. J. Stat. 40 (1), 137-150, 2024.
• [27] S. Särkkä and A. Solin, Applied Stochastic Differential Equations, Cambridge Univ. Press, 2019.
• [28] J. G. Hayes and E. J. Allen, Stochastic point-kinetics equations in nuclear reactor dynamics, Ann. Nucl. Energy 2, 572-587, 2005.
• [29] M. J. Panik, Stochastic Differential Equations: An Introduction with Applications in Population Dynamics Modeling, John Wiley & Sons, Hoboken, NJ, 2017.
• [30] T. Mikosch, Elementary Stochastic Calculus, with Finance in View, World Scientific, 1998.
• [31] S. M. Ross, An Introduction to Mathematical Finance, Cambridge Univ. Press, 1999.
• [32] R. Korn and E. Korn, Option Pricing and Portfolio Optimization: Modern Methods of Financial Mathematics, Amer. Math. Soc. 31, 2001.
• [33] D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev. 43, 525-546, 2001.
• [34] S. M. Iacus, Simulation and Inference for Stochastic Differential Equations: With R Examples, Springer, New York, 486, 2008.
• [35] M. Bayram, T. Partal and G. O. Buyukoz, Numerical methods for simulation of stochastic differential equations, Adv. Differ. Equ. 2018 (1), 1-10, 2018.
• [36] M. Kessler, A. Lindner and M. Sørensen, Statistical Methods for Stochastic Differential Equations, Chapman & Hall/CRC, 2019.
• [37] S. S. Shapiro and M. B. Wilk, An analysis of variance test for normality (complete samples), Biometrika 52 (3/4), 591-611, 1965.
• [38] D. Kwiatkowski, P. C. Phillips, P. Schmidt and Y. Shin, Testing the null hypothesis of stationarity against the alternative of a unit root: How sure are we that economic time series have a unit root?, J. Econometrics 54 (1-3), 159-178, 1992.
• [39] G. E. P. Box and D. A. Pierce, Distribution of residual autocorrelation in autoregressive-integrated moving average time series models, J. Amer. Stat. Assoc. 65, 1509-1526, 1970.